Accelerated finite difference schemes for second order degenerate elliptic and parabolic problems in the whole space
aa r X i v : . [ m a t h . A P ] M a y ACCELERATED FINITE DIFFERENCE SCHEMES FORSECOND ORDER DEGENERATE ELLIPTIC ANDPARABOLIC PROBLEMS IN THE WHOLE SPACE
ISTV ´AN GY ¨ONGY AND NICOLAI KRYLOV
Abstract.
We give sufficient conditions under which the convergenceof finite difference approximations in the space variable of possibly de-generate second order parabolic and elliptic equations can be acceleratedto any given order of convergence by Richardson’s method. Introduction
This is the third article of a series studying a class of finite differenceequations, related to finite difference approximations in the space variable of second order parabolic and elliptic PDEs in R d . These PDEs are givenon the whole R d in the space variable, and may degenerate and become firstorder PDEs. Denote by u h the solutions of the finite difference equationscorresponding to a given grid with mesh-size h . By shifting the grid so that x becomes a grid point we define u h for all x ∈ R d rather than only at thepoints of the original grid. In [5] and [6], the first and second articles of theseries, we focus on the smoothness in x of u h , rather than their convergencefor h →
0. The main results in [5] and [6] give estimates, independent of h , for the first order derivatives Du h and for derivatives D k u h in x of anyorder k , respectively.In the present paper one of our main concerns is the smoothness of theapproximations u h in ( x, h ). In particular, we are interested in the conver-gence of u h , and their derivatives in x , in the supremum norm, as h →
0. Wegive conditions ensuring that for any given integer k ≥ u h admit power series expansions up to order k + 1 in h near 0 like u h = k X j =0 h j u ( j ) + h k +1 r h , (1.1)and such that the coefficients are bounded functions of ( t, x ) ∈ [0 , T ] × R d for fixed T > r h , are independent of h . This is Theorem 2.3, our first result on Taylor’s Mathematics Subject Classification.
Key words and phrases.
Cauchy problem, finite differences, extrapolation to the limit,Richardson’s method .The work of the second author was partially supported by NSF grant DMS-0653121. formula for u h in h . We obtain it by proving first Theorems 2.1 and 2.2below on the solvability of the PDE that is being approximated, and of asystem of degenerate parabolic PDEs, respectively, for the coefficients u ( j ) , j = 0 , . . . , k . Of course, u (0) is the true solution of the corresponding PDE.The remainder term r h satisfies a finite difference equation, with the samedifference operator appearing in the equation for u h , and we estimate r h by making use of the maximum principle enjoyed by this operator. Thisis a standard approach to get power series expansions for finite differenceapproximations in general, and it works well in many situations, when suit-able results regarding the equations for the coefficients u ( j ) are available.In our situation it requires some facts either from the theory of diffusionprocesses or from the theory of degenerate parabolic equations. However,we do not use any facts from these theories. We prove Theorem 2.1, andhence Theorem 2.2, relying on results on finite difference schemes, obtainedin [6] by elementary techniques. It is worth saying that since long ago finitedifference equations were already used to prove the solvability of partial dif-ferential equations (see, for instance, [8] and [9]). Our contribution lies inconsidering degenerate equations.After establishing the expansions of u h in h not only we obtain the pos-sibility to prove the convergence of u h to the true solution in the sup normas h → u h are zero. These results are given by Theorem 2.21 and Corollary 2.8.Their counterparts in the elliptic case are presented by Corollary 3.7.The idea of accelerating the convergence of finite difference approxima-tions in the above way is well-known in numerical analysis. It is due toL.F. Richardson, who showed that it works in some cases and demonstratedits usefulness in [15] and [16]. This method is often called Richardson’smethod or extrapolation to the limit , and is applied to various types of ap-proximations. The reader is referred to the survey papers [2] and [4] for areview on the history of the method and on the scope of its applicabilityand to the textbooks (for instance, [10] and [11]) concerning finite differencemethods and their accelerations.We are interested in approximating in the sup norm not only the truesolution but also its derivatives. Note that even if the coefficients u ( j ) arebounded smooth functions of ( t, x ), the derivatives D k u h of u h in x may notadmit similar expansions, since the derivatives of r h may not be boundedin h near 0. Note also that the bounds on the sups of u ( j ) and r h generallydepend on T , and may grow exponentially in T . This becomes a big obstacleon the way of extending our results to the elliptic case.Our next result on power series expansions, Theorem 2.7, improves theprevious theorem in two directions. It gives sufficient conditions such thatfor any given integer k ≥ CCELERATED SCHEMES 3 (a) D k u h admits an expansion similar to (1.1),(b) the bounds on the coefficients are independent of T .Having (a) we can approximate the k -th derivatives of the true solutionby D k u h with rate of order h and accelerate the rate under appropriateassumptions. We can also approximate the k -th derivatives of the truesolution with finite difference operators in place of D k applied to u h , whichis more convenient in applications because it does not require computingthe derivatives of u h .We ensure (a) and (b) by relying heavily on derivative estimates, indepen-dent of T , obtained in [5] and [6] for solutions of finite difference equations.Property (b) of the expansions allows us to extend Theorem 2.7 to the el-liptic case. This extension is Theorem 3.5.As a consequence of the derivative estimates proved in [6] we obtain also,see Theorem 2.9 below, estimates, independent of h and T , for the deriva-tives of u h in x and h . Clearly, Theorem 2.9 immediately implies Taylor’sformula for u h in h , up to appropriate order, with bounded coefficients. Itis interesting to notice that the converse implication does not hold: If for k ≥ u h admits a power series expansion up to order k + 1 in h near 0 with bounded coefficients, it does not imply, in general, that thederivative of u h in h up to order k + 1 are bounded functions. That is whyTheorem 2.7 does not imply Theorem 2.9, and the latter implies the formeronly if condition (i) in Theorem 2.7 is satisfied. Additional information onthe behaviour of the derivatives of u h in x and h when h is near 0 is given byTheorem 2.11. The corresponding result in the elliptic case is Theorem 3.4.In this article we are working with equations in the whole space havingin mind considering equations in bounded smooth domains in a subsequentarticle. Still it may be worth noting that the results of this article areapplicable to the one dimensional ODE(1 − x ) u ′′ ( x ) − c ( x ) u ( x ) = f ( x ) , x ∈ ( − , . The point is that one need not prescribe any boundary value of u at thepoints ± R , the values of itscoefficients and f outside ( − ,
1) do not affect the values of u ( x ) for | x | < Formulation of the main results for parabolic equations
We fix some numbers h , T ∈ (0 , ∞ ) and for each number h ∈ (0 , h ] weconsider the integral equation u ( t, x ) = g ( x ) + Z t (cid:0) L h u ( s, x ) + f ( s, x ) (cid:1) ds, ( t, x ) ∈ H T (2.1)for u , where g ( x ) and f ( s, x ) are given real-valued Borel functions of x ∈ R d and ( s, x ) ∈ H T = [0 , T ] × R d , respectively, and L h is a linear operatordefined by L h ϕ ( t, x ) = L h ϕ ( t, x ) − c ( t, x ) ϕ ( x ) , (2.2) I. GY ¨ONGY AND N. KRYLOV L h ϕ ( t, x ) = 1 h X λ ∈ Λ q λ ( t, x ) δ h,λ ϕ ( x ) + X λ ∈ Λ p λ ( t, x ) δ h,λ ϕ ( x ) , (2.3)for functions ϕ on R d . Here Λ is a finite subset of R d such that 0 Λ , δ h,λ ϕ ( x ) = 1 h ( ϕ ( x + hλ ) − ϕ ( x )) , λ ∈ Λ ,q λ ( t, x ) ≥ p λ ( t, x ), and c ( t, x ) are given real-valued Borel functions of( t, x ) ∈ H ∞ = [0 , ∞ ) × R d for each λ ∈ Λ . Set | Λ | = P λ ∈ Λ | λ | .As usual, we denote D α = D α ...D α d d , D i = ∂∂x i , | α | = X i α i , D ij = D i D j for multi-indices α = ( α , . . . α d ), α i ∈ { , , . . . } . For smooth ϕ and integers k ≥ D k ϕ as the collection of partial derivatives of ϕ of order k , and define | D k ϕ | = X | α | = k | D α ϕ | , [ ϕ ] k = sup x ∈ R d | D k ϕ ( x ) | , | ϕ | k = X i ≤ k [ ϕ ] i . For functions ψ h depending on h ∈ (0 , h ] the notation D kh ψ h means the k -th derivative of ψ in h . For Borel measurable bounded functions ψ = ψ ( t, x )on H T we write ψ ∈ B m = B mT if, for each t ∈ [0 , T ], ψ ( t, x ) is continuousin R d and for all multi-indices α with | α | ≤ m the generalized functions D α ψ ( t, x ) are bounded on H T . In this case we use the notation k ψ k m = sup H T X | α |≤ m | D α ψ ( t, x ) | . This notation will be also used for functions ψ independent of t .Let m ≥ Assumption 2.1.
For any λ ∈ Λ , we have p λ , q λ , c, f, g ∈ B m and, for k = 0 , ..., m and some constants M k we havesup H T (cid:0) X λ ∈ Λ ( | D k q λ | + | D k p λ | (cid:1) + | D k c | (cid:1) ≤ M k . (2.4) Remark . By Theorem 2.3 of [5] under Assumption 2.1 for each h ∈ (0 , h ], there exists a unique bounded solution u h of (2.1), this solution iscontinuous in H T , and all its derivatives in x up to order m are bounded.Actually, in Theorem 2.3 of [5] it is required that the derivatives of the dataup to order m be continuous in H T , but its proof can be easily adjusted toinclude our case (see Remark 2.6 below).Naturally, we view (2.1) as a finite difference schemes for the problem ∂∂t u ( t, x ) = L u ( t, x ) + f ( t, x ) , t ∈ (0 , T ] , x ∈ R d , (2.5) u (0 , x ) = g ( x ) , x ∈ R d , (2.6) CCELERATED SCHEMES 5 where L := X λ ∈ Λ d X i,j =1 q λ λ i λ j D i D j + X λ ∈ Λ d X i =1 p λ λ i D i − c. (2.7)By a solution of (2.5)-(2.6) we mean a bounded continuous function u ( t, x )on H T , such that it belongs to B and satisfies u ( t, x ) = g ( x ) + Z t [ L u ( s, x ) + f ( s, x )] ds (2.8)in H T in the sense of generalized functions, that is, for any t ∈ [0 , T ] and φ ∈ C ∞ ( R d ) Z R d φ ( x ) u ( t, x ) dx = Z R d φ ( x ) g ( x ) dx + Z t Z R d φ ( − cu + f )( s, x ) dxds + Z t Z R d φ X λ ∈ Λ (cid:0) d X i,j =1 q λ λ i λ j D i D j u + d X i =1 p λ λ i D i u (cid:1) ( s, x ) dxds. (2.9)Observe that if u ∈ B , then (2.9) implies that (2.8) holds almost everywherewith respect to x and if u ∈ B then the second derivatives of u in x arecontinuous in x and (2.8) holds everywhere.The reader can find in [7] a discussion showing that in all practicallyinteresting cases of parabolic equations like (2.8) the operator L can berepresented as in (2.7), so that considering operators L h in form (2.3) israther realistic.The following theorem on existence and uniqueness of solutions is a clas-sical result (see, for instance, [12], [13], [14]) which we are going to obtainby using finite-difference approximations. Theorem 2.1.
Let Assumption 2.1 hold with m ≥ . Then equation (2.8) has a unique solution u (0) ∈ B = B T . Moreover, u (0) ∈ B mT and k u (0) k m ≤ N ( k f k m + k g k m ) , (2.10) where N is a constant, depending only on d , m , | Λ | , M ,. . . , M m , and T . Observe that this result is rather sharp in what concerns the smoothnessof solutions, which is seen if all the coefficients of L are identically zero and f is independent of t in which case the solution is tf ( x ) + g ( x ).The existence part in Theorem 2.1 is proved in Section 6 and uniquenessin Section 4.In Section 6 a repeated application of this theorem allows us to prove aresult on the solvability of (2.13) below. First introduce L ( i ) := i +1)( i +2) X λ ∈ Λ q λ ∂ i +2 λ + i +1 X λ ∈ Λ p λ ∂ i +1 λ , (2.11)where ∂ λ ϕ := X i λ i D i ϕ (2.12) I. GY ¨ONGY AND N. KRYLOV is the derivative of ϕ in the direction of λ . Consider the system of equations u ( j ) ( t, x ) = Z t (cid:0) L u ( j ) ( s, x ) + j X i =1 C ij L ( i ) u ( j − i ) ( s, x ) (cid:1) ds, (2.13)( t, x ) ∈ H T , j = 1 , . . . , k . Remark . Quite often in the article we use the following symmetry con-dition:(S) Λ = − Λ and q λ = q − λ for all λ ∈ Λ .Notice that, if condition (S) holds, then h − X λ ∈ Λ q λ ( t, x ) δ h,λ ϕ ( x ) = (1 / X λ ∈ Λ q λ ( t, x )∆ h,λ ϕ ( x ) , where ∆ h,λ ϕ ( x ) = h − ( ϕ ( x + hλ ) − ϕ ( x ) + ϕ ( x − hλ )) . Theorem 2.2.
Let k ≥ be an integer. (i) If Assumption 2.1 is satisfiedwith m ≥ k + 2 , then (2.13) has a unique solution { u ( j ) } kj =1 , such that u ( j ) ∈ B m − j , k u ( j ) k m − j ≤ N ( k f k m + k g k m ) (2.14) for j = 1 , . . . , k .(ii) If the symmetry condition (S) holds and Assumption 2.1 is satisfiedwith m ≥ k + 2 , then (2.13) has a unique solution { u ( j ) } kj =1 , such that u ( j ) ∈ B m − j , k u ( j ) k m − j ≤ N ( k f k m + k g k m ) (2.15) for j = 1 . . . , k . In addition, if p − λ = − p λ , for λ ∈ Λ , (2.16) then u ( j ) = 0 , (2.17) for odd numbers j ≤ k .In all cases the constants N depends only on d , m , | Λ | , M , . . . , M m ,and T . The next series of results is related to the possibility of expansion u h ( t, x ) = u (0) ( t, x ) + X ≤ j ≤ k h j j ! u ( j ) ( t, x ) + h k +1 r h ( t, x ) , (2.18)for all ( t, x ) ∈ H T and h ∈ (0 , h ], where u h is the unique bounded solution of(2.1) (see Remark 2.1) and r h is a function on H T defined for each h ∈ (0 , h ]such that | r h ( t, x ) | ≤ N ( k f k m + k g k m ) (2.19)for all ( t, x ) ∈ H T , h ∈ (0 , h ].Introduce χ h,λ = q λ + hp λ . CCELERATED SCHEMES 7
Assumption 2.2.
For all ( t, x ) ∈ H T , h ∈ (0 , h ], and λ ∈ Λ , χ h,λ ( t, x ) ≥ . (2.20) Assumption 2.3.
We have X λ ∈ Λ λq λ ( t, x ) = 0 for all ( t, x ) ∈ H T . Notice that condition (S) is stronger than Assumption 2.3.
Theorem 2.3.
Let Assumption 2.1 with m ≥ and Assumption 2.2 hold.Let k ≥ be an integer. Then expansion (2.18) holds with r h satisfying (2.19) , provided one of the following conditions is met:(i) m ≥ k + 3 and Assumption 2.3 holds;(ii) m ≥ k + 3 and condition (S) holds;(iii) k is odd, m ≥ k + 2 , and conditions (S) and (2.16) are satisfied.In each of the cases (i)-(iii) the constant N depends only on d , m , | Λ | , M , . . . , M m , and T . In case (iii) we have u ( j ) = 0 for all odd j in expansion (2.18) . We prove this theorem in Section 7. The following corollary is one of theresults of [3] proved there by using the theory of diffusion processes. Weobtain it immediately from case (iii) with k = 1. Of course, the result iswell known for uniformly nondegenerate equations but we do not assumeany nondegeneracy of L , which becomes just a zero operator at those pointswhere q λ = p λ = c = 0. Corollary 2.4.
Let conditions (S) and (2.16) be satisfied. Let Assumption2.1 with m = 4 and Assumption 2.2 hold. Then we have | u h − u | ≤ N h . Actually, in [3] a full discretization in time and space is considered forparabolic equations, so that, formally, Corollary 2.4 does not yield the cor-responding result of [3]. On the other hand, a similar corollary can bederived from Theorem 3.5 below which treats elliptic equations and it doesimply the corresponding result of [3]. It also generalizes it because in [3] oneof the assumptions, unavoidable for the methods used there, is that q λ = r λ with functions r λ that have four bounded derivatives in x , which may easilybe not the case under the assumptions of Theorem 3.5.To formulate our main result about acceleration for parabolic equationswe fix an integer k ≥ u h = k X j =0 b j u − j h , , (2.21)where, naturally, u − j h are the solutions to (2.1), with 2 − j h in place of h ,( b , b , ..., b k ) := (1 , , , ..., V − (2.22)and V − is the inverse of the Vandermonde matrix with entries V ij := 2 − ( i − j − , i, j = 1 , ..., k + 1 . I. GY ¨ONGY AND N. KRYLOV
The following result is a simple corollary of Theorem 2.3.
Theorem 2.5.
In each situation when Theorem 2.3 is applicable we havethat the estimate | ¯ u h ( t, x ) − u (0) ( t, x ) | ≤ N ( k f k m + k g k m ) h k +1 (2.23) holds for all ( t, x ) ∈ H T , h ∈ (0 , h ] , where N is a constant depending onlyon d , m , | Λ | , M , . . . , M m , and T .Proof. By Theorem 2.3 u − j h = u (0) + k X i =1 h i i !2 ji u ( i ) + ¯ r − j h h k +1 , j = 0 , , ..., k, with ¯ r − j h := 2 − j ( k +1) r − j h , which gives¯ u h = k X j =0 b j u − j h = ( k X j =0 b j ) u (0) + k X j =0 k X i =1 b j h i i !2 ij u ( i ) + k X j =0 b j ¯ r − j h h k +1 = u (0) + k X i =1 h i i ! u ( i ) k X j =0 b j ij + k X j =0 b j ¯ r − j h = u (0) + k X j =0 b j ¯ r − j h h k +1 , since k X j =0 b j = 1 , k X j =0 b j − ij = 0 , i = 1 , , ...k by the definition of ( b , ..., b k ). Hence,sup H T | ¯ u h − u (0) | = sup H T | k X j =0 b j ¯ r − j h | h k +1 ≤ N ( k f k m + k g k m ) h k +1 , and the theorem is proved. (cid:3) Sometimes it suffices to combine fewer terms u − j h to get accuracy oforder k + 1. To consider such a case for odd integers k ≥ u h = ˜ k X j =0 ˜ b j u − j h , (2.24)where (˜ b , ˜ b , ..., ˜ b ˜ k ) := (1 , , , ...,
0) ˜ V − , ˜ k = k − , (2.25)and ˜ V − is the inverse of the Vandermonde matrix with entries˜ V ij := 4 − ( i − j − , i, j = 1 , ..., ˜ k + 1 . Theorem 2.6.
Suppose that the assumptions of Theorem 2.3 are satisfiedand condition (iii) is met. Then for ˜ u h we have sup H T | u (0) − ˜ u h | ≤ N ( k f k m + k g k m ) h k +1 for all h ∈ (0 , h ] , where N depends only on d , m , | Λ | , M , . . . , M m , and T . CCELERATED SCHEMES 9
Proof.
We obtain this result from Theorem 2.3 by a straightforward modi-fication of the proof of the previous result, taking into account that for odd j the terms with h j vanish in expansion (2.18) when condition (iii) holds inTheorem 2.3. (cid:3) Example 2.1.
Assume that in the situation of Theorem 2.6 we have m = 8.Then ˜ u h := u h/ − u h satisfies sup H T | u (0) − ˜ u h | ≤ N h for all h ∈ (0 , h ].The above results show that if the data in equation (2.8) are sufficientlysmooth, then the order of accuracy in approximating the solution u (0) canbe as high as we wish if we use suitable mixtures of finite difference approx-imations calculated along nested grids with different mesh-sizes. Assumenow that we need to approximate not only u (0) but its derivative D α u (0) forsome multi-index α as well. What accuracy can we achieve? The answer isclosely related to the question whether the expansion D α u h ( t, x ) = D α u (0) ( t, x ) + X ≤ j ≤ k h j j ! D α u ( j ) ( t, x ) + h k +1 D α r h ( t, x ) (2.26)holds for all ( t, x ) ∈ H T and h ∈ (0 , h ], such that | D α r h ( t, x ) | ≤ N ( k f k m + k g k m ) (2.27)for all ( t, x ) ∈ H T , h ∈ (0 , h ].The result concerning this expansion and the following series of resultsappeared after the authors tried to extend the above theorems from theparabolic to the elliptic case. The main and rather hard obstacle is that theconstants in our estimates depend on T and, actually, may grow exponen-tially in T . By the way, this obstacle is caused by possible degeneration ofour equations and exists even if we consider equations in bounded smoothdomain.To be able to give some conditions under which this does not happen,we introduce new notation and investigate smoothness properties of u h withrespect to x . As a simple byproduct of this investigation we also obtainsmoothness of u h with respect to h , which, by the way, cannot be derivedfrom (2.18).Take a function τ λ defined on Λ taking values in [0 , ∞ ) and for λ ∈ Λ introduce the operators T h,λ ϕ ( x ) = ϕ ( x + hλ ) , ¯ δ h,λ = τ λ h − ( T h,λ − . Set k Λ k = X λ ∈ Λ | τ λ λ | . For uniformity of notation we also introduce Λ as the set of fixed distinctvectors ℓ , ..., ℓ d none of which is in Λ and define¯ δ h,ℓ i = τ D i , T h,ℓ i = 1 , Λ = Λ ∪ Λ , where τ > λ = ( λ , λ ) ∈ Λ introduce theoperators T h,λ = T h,λ T h,λ , ¯ δ h,λ = ¯ δ h,λ ¯ δ h,λ . For k = 1 , µ ∈ Λ k we set Q h,µ ϕ = h − X λ ∈ Λ (¯ δ h,µ q λ ) δ λ ϕ, L h,µ ϕ = Q h,µ ϕ + X λ ∈ Λ (¯ δ h,µ p λ ) δ λ ϕ,A h ( ϕ ) = 2 X λ ∈ Λ (¯ δ h,λ ϕ ) L h,λ T h,λ ϕ, Q h ( ϕ ) = X λ ∈ Λ χ h,λ ( δ h,λ ϕ ) . Below B ( R d ) is the set of bounded Borel functions on R d and K is the setof bounded operators K h = K h ( t ) mapping B ( R d ) into itself preserving thecone of nonnegative functions and satisfying K h ≤ δ ∈ (0 ,
1] and K ∈ [1 , ∞ ). Assumption 2.4.
There exists a constant c > c ≥ c . Remark . The above assumption is almost irrelevant if we only consider(2.1) on a finite time interval. Indeed, if c is just bounded, say | c | ≤ C =const, by introducing a new function v ( t, x ) = u ( t, x ) e − Ct we will have anequation for v similar to (2.1) with L h v − ( c + 2 C ) v and f e − Ct in place of L h u and f , respectively. Now for the new c we have c + 2 C ≥ C . Assumption 2.5.
We have m ≥ h ∈ (0 , h ], there exists anoperator K h = K h,m ∈ K , such that mA h ( ϕ ) ≤ (1 − δ ) X λ ∈ Λ Q h (¯ δ h,λ ϕ ) + K Q h ( ϕ ) + 2(1 − δ ) c K h (cid:0) X λ ∈ Λ | ¯ δ h,λ ϕ | (cid:1) (2.28)on H T for all smooth functions ϕ . Assumption 2.6.
We have m ≥ h ∈ (0 , h ] and n = 1 , ..., m ,there exists an operator K h = K h,n ∈ K , such that n X ν ∈ Λ A h (¯ δ h,ν ϕ ) + n ( n − X λ ∈ Λ (¯ δ h,λ ϕ ) Q h,λ T h,λ ϕ ≤ (1 − δ ) X λ ∈ Λ Q h (¯ δ h,λ ϕ )+ K X λ ∈ Λ Q h (¯ δ hλ ϕ ) + 2(1 − δ ) c K h (cid:0) X λ ∈ Λ | ¯ δ h,λ ϕ | (cid:1) + K K (cid:0) X λ ∈ Λ | ¯ δ h,λ ϕ | (cid:1) (2.29)on H T for all smooth functions ϕ .Obviously Assumptions 2.5 and 2.6 are satisfied if q λ and p λ are inde-pendent of x . In the general case, as it is discussed in [5], the above as-sumptions impose not only analytical conditions, but they are related alsoto some structural conditions, which can somewhat easier be analized underthe symmetry condition (S). CCELERATED SCHEMES 11
Assumption 2.7.
For all t ∈ [0 , T ] X λ ∈ Λ λq λ ( t, x ) is independent of x. (2.30)In the main case of applications we will require the last sum to be iden-tically zero as in Assumption 2.3. Remark . Assumptions 2.5 and 2.6 are discussed at length and in manydetails in [5] and [6], and sufficient conditions, without involving test func-tions ϕ are given for these assumptions to be satisfied. In particular, it isshown in [6] that if condition (S) holds, m ≥ τ λ = 1, Assumptions 2.1 and2.2 are satisfied, and q λ ≥ κ for a constant κ >
0, then both Assumptions 2.5and 2.6 are satisfied for any c > δ ∈ (0 , h is sufficiently smalland τ , K , and K are chosen appropriately. Moreover, the condition κ > c is large enough (this timewe need not assume that h is small). Remember, that by Remark 2.3 thecondition that c be large is, actually, harmless as long as we are concernedwith equations on a finite time interval. Mixed situations, when c is largeat those points where some of q λ can vanish are also considered in [6].In [5] we have seen that Assumption 2.5 imposes certain nontrivial struc-tural conditions on q λ which cannot be guaranteed by the size of c if q λ isonly once continuously differentiable. In contrast, even without condition(S), given that Assumptions 2.1, 2.5, 2.7 are satisfied and m ≥
2, as is shownin [6], Assumption 2.6 is also satisfied if c is large enough. Theorem 2.7.
Let Assumption 2.1 through 2.6 hold with m ≥ . Let k ≥ and l ∈ [0 , m ] be integers. Then for every multi-index α such that | α | ≤ l thefunction D α u h is a continuous function on H T and expansion (2.26) holdswith D α r h satisfying (2.27) , provided one of the following conditions is met:(i) m ≥ k + 3 + l ;(ii) m ≥ k + 3 + l and condition (S) holds;(iii) k is odd, m ≥ k + 2 + l , and conditions (S) and (2.16) are satisfied.In each of the cases (i)-(iii) the constant N depends only on d , m , δ , K , τ , c , | Λ | , k Λ k , M , . . . , M m . In case (iii) we have u ( j ) = 0 for all odd j inthe expansion. We prove this theorem in Section 7. Remember the definition of ¯ u h and ˜ u h in (2.21) and (2.24). The following is an obvious consequence of Theorem 2.7. Corollary 2.8.
Suppose that the assumptions of Theorem 2.7 are satisfied.Then sup H T | D α ¯ u h − D α u (0) | ≤ N h k +1 ( k f k m + k g k m ) , and if condition (iii) is met then sup H T | D α ˜ u h − D α u (0) | ≤ N h k +1 ( k f k m + k g k m ) , where N depends only on on d , m , δ , K , τ , c , | Λ | , k Λ k , M , . . . , M m . Remark . Observe that for k = 0 Theorem 2.7 implies thatsup H T | D α u h − D α u (0) | ≤ N h (2.31)if m ≥ | α | and Assumption 2.1 through 2.6 hold. In addition one canreplace D α u h in (2.31) with δ αh , where δ αh = δ α h,e · ... · δ α d h,e d and e i is the i th basis vector in R d . This follows easily from the meanvalue theorem and Theorem 2.9 below. The reader understands that similarassertion is true in case of Corollary 2.8 with the only difference that oneneeds larger m and better finite-difference approximations of D α .Next we investigate the smoothness of u h in x and h . Recall that forfunctions ϕ depending on h we use the notation D rh ϕ for the r -th derivativeof ϕ in h . As usual, D h ϕ := ϕ . Remark . Suppose that Assumption 2.1 is satisfied. Take an h ∈ (0 , h ),consider equation (2.1) as an equation about a function u h ( t, x ) as function of( h, t, x ) ∈ [ h , h ] × H T and look for solutions in the space B m ( h ) = B mT ( h )which is defined as the space of functions on [ h , h ] × H T with finite norm X | α | +3 r ≤ m sup [ h ,h ] × H T | D α D rh u h ( t, x ) | . (2.32)It is obvious that the integrand in (2.1) can be considered as the resultof application of an operator, which is bounded in B m ( h ), to u h ( s, x ).Therefore, a standard abstract theorem on solvability of ODEs in Banachspaces shows that there exists a solution of (2.1) in B m ( h ). Since justbounded solutions are uniquely defined by (2.1), we conclude that our u h belongs to B m ( h ) for any h ∈ (0 , h ). Obviously, if the derivatives of thedata are continuous in x , the same will hold for u h .The above argument, actually, works if we replace | α | + 3 r ≤ m with | α | + r ≤ m in (2.32). We talk about (2.32) in the above form because wewill show that under our future assumptions the quantity (2.32) is boundedindependently of h . Theorem 2.9.
Let k ≥ and m ≥ be integers and suppose that As-sumptions 2.1 through 2.6 are satisfied. Then, for each integer r ≥ suchthat k + r ≤ m, the generalized derivatives D r D kh u h exist on (0 , h ] × H T , are bounded andwe have | D r D kh u h | ≤ N ( k f k m + k g k m ) , (2.33) where N is a constant depending only on m, δ, c , τ , K , M , ..., M m , | Λ | ,and k Λ k . In particular, u h ∈ B m and k u h k m ≤ N ( k f k m + k g k m ) . CCELERATED SCHEMES 13
We prove this theorem in Section 5, and in Section 6 we show that thefollowing fact, used when we come to the elliptic case, is a simple corollaryof it.
Theorem 2.10.
Suppose that Assumptions 2.1 through 2.6 hold with m ≥ . Then the constant N in (2.10) depends only on m, δ, c , τ , K , M , ..., M m , | Λ | , and k Λ k (thus, is independent of T ). The same is true for theconstants N in Theorems 2.2, 2.3, 2.5, and 2.6. Additional information on the behavior of D r D kh u h for small h is providedby the following result which we prove in Section 5. Theorem 2.11.
Let k ≥ be an odd number and suppose that Assumptions2.1 through 2.6 hold with m ≥ k + 1 . Assume that the symmetry condition(S) and (2.16) are satisfied.Then, for any integer r ≥ such that k + r ≤ m − we have sup H T | D r D kh u h | ≤ N ( k f k m + k g k m ) h (2.34) for all h ∈ (0 , h ] , where N depends only on m , δ , c , τ , K , | Λ | , k Λ k , M ,..., M m . Main results for elliptic equations
Here we assume that p λ , q λ , c , and f are independent of t and turn nowour attention to the equations L h v h ( x ) + f ( x ) = 0 x ∈ R d , (3.1) L v ( x ) + f ( x ) = 0 x ∈ R d . (3.2)Naturally by a solution of (3.2) we mean a function v on R d such thatit belongs to B and (3.2) holds almost everywhere. Clearly, if a solution v belongs to B and q λ , p λ , c , and f are continuous functions on R d , then(3.2) holds everywhere.First we prove the existence and uniqueness of the solutions of equations(3.1) and (3.2). Theorem 3.1.
Suppose that Assumption 2.1 is satisfied with an m ≥ andlet Assumptions 2.2 and 2.4 hold. Then equation (3.1) has a unique boundedsolution v h . Moreover, v h belongs to B m .Proof. Observe that (3.1) is equivalent to v h ( x ) = h ξ ( x ) f ( x ) + ξ ( x ) X λ ∈ Λ χ λ v h ( x + λh ) , where ξ − = h c + X λ ∈ Λ χ λ . It is seen that the existence and uniqueness of bounded solution of (3.1)follows by contraction principle. Using smooth successive iterations yieldsthat v h ∈ B m . (cid:3) Theorem 3.2.
Let Assumptions 2.1 through 2.6 hold with an m ≥ . Thenequation (3.2) has a unique solution v in the space B . Moreover, v ∈ B m and there is a constant N depending only on m , δ , c , τ , K , M , ..., M m , | Λ | , and k Λ k such that k v k m ≤ N k f k m . (3.3) Proof.
First we prove uniqueness. Let v ∈ B satisfy (3.2) with f = 0. Takea constant ν >
0, so small that c − ν ≥ c / c − ν and δ/ c and δ , respectively. Then for each T > u ( t, x ) := e νt v ( x )), ( t, x ) ∈ H T , is a solution of class B T of theequation ∂∂t u = ( L + ν ) u on H T (3.4)with initial condition u (0 , x ) = v ( x ). Hence by virtue of Theorem 2.10 forevery T > e νT | v ( x ) | = | u ( T, x ) | ≤ N k v k , where N is independent of ( T, x ). Multiplying both sides of the above in-equality by e − νT and letting T → ∞ we get v = 0, which proves uniqueness.To show the existence of a solution in B m , let u be a function defined on H ∞ such that for each T > H T is the unique solutionin B mT of (3.4) with initial condition u (0 , x ) = f ( x ) (see Theorem 2.1). ByTheorem 2.10 sup H ∞ X r ≤ m | D r u | ≤ N k f k m with a constant N depending only on m , δ , c , τ , K , M , ..., M m , | Λ | , and k Λ k . Hence v ( x ) := Z ∞ e − νt u ( t, x ) dt, x ∈ R d is a well-defined function on R d , v ∈ B m , and L v ( x ) = Z ∞ e − νt L u ( t, x ) dt = Z ∞ e − νt ( ∂∂t u ( t, x ) − νu ( t, x )) dt = − f ( x ) , where the last equality is obtained by integration by parts. Consequently, v is a solution of (3.4) and it satisfies estimate (3.3). (cid:3) Theorem 3.3.
Let k ≥ and suppose that Assumptions 2.1 through 2.6are satisfied with an m ≥ k . Then, for any h ∈ (0 , h ] and for each integer r ≥ , such that k + r ≤ m, CCELERATED SCHEMES 15 for the unique bounded solution v h of (3.1) we have sup (0 ,h ] × R d | D r D kh v h | ≤ N k f k m , (3.5) where N is a constant depending only on m, δ, c , τ , K , | Λ | , k Λ k , M , ..., M m . In particular, k v h k m ≤ N k f k m . Proof.
To prove (3.5), take a constant ν > u ( t, x ) := v h ( x ) e νt , and observe that u is the unique bounded solutionof ∂∂t u = L h u − ( c − ν ) u + e νt f, u (0 , x ) = v h ( x ) . By Theorem 2.9 for any
T > e νT | D r D kh v h ( x ) | = | D r D kh u ( T, x ) | ≤ N e νT k f k m + N k v h k m , where N is a constant, depending only on m, δ, c , τ , K , | Λ | , k Λ k , M , ..., M m . By multiplying the extreme terms by e − νT and letting T → ∞ , we getthe result. (cid:3) From estimate (2.34) we obtain the corresponding estimate for the deriva-tives of v h . Theorem 3.4.
Let the conditions of Theorem 2.11 hold. Then for anyinteger r ≥ such that k + r ≤ m − , for the solution v h of (3.1) we have sup R d | D r D kh v h | ≤ N k f k m h for all h ∈ (0 , h ] , where N depends only on m, δ, c , τ , K , | Λ | , k Λ k and M ,..., M m .Proof. This theorem can be deduced from Theorem 2.11 in the same way asTheorem 3.3 is obtained from Theorem 2.9. (cid:3)
Now we want to establish an expansion for v h , i.e., to show for an integer k ≥ v (0) ,..., v ( k ) on R d , and a function R h on R d for each h ∈ (0 , h ] such that for all x ∈ R d and h ∈ (0 , h ] v h ( x ) = v (0) ( x ) + X ≤ j ≤ k h j j ! v ( j ) ( x ) + h k +1 R h ( x ) , (3.6)sup h ∈ (0 ,h ] sup R d | R h | ≤ N k f k m (3.7)with a constant N . Theorem 3.5.
Suppose that Assumptions 2.1 through 2.6 are satisfied withan m ≥ . Let k ≥ be an integer. Then expansion (3.6) holds with v (0) being the unique B m solution of (3.2) and R h satisfying (3.7) provided oneof the following conditions is met:(i) m ≥ k + 3 ;(ii) m ≥ k + 3 and condition (S) holds;(iii) k is odd, m ≥ k + 2 , and conditions (S) and (2.16) are satisfied.In each of the cases (i)-(iii) the constant N in (3.7) depends only on d , m , δ, c , τ , K , | Λ | , k Λ k , M , . . . , M m . Moreover, when (iii) holds we have v ( j ) = 0 for all odd j .Proof. Take a small constant ν >
0, as in the proof of Theorem 3.2, let u bea function defined on H ∞ such that for each T > H T is the unique solution in B mT of ∂∂t u h = ( L h + ν ) u h ( t, x ) ∈ H ∞ u h (0 , x ) = f ( x ) x ∈ R d , (see Remark 2.1). As in the proof of Theorem 3.2 we get that v h ( x ) = Z ∞ e − νt u h ( t, x ) dt. By Theorem 2.3 in each of the cases (i)-(iii) we have u h ( t, x ) = u (0) ( t, x ) + X ≤ j ≤ k h j j ! u ( j ) ( t, x ) + h k +1 r h ( t, x ) , (3.8)for all ( t, x ) ∈ H ∞ , h ∈ (0 , h ], and by Theorem 2.10 we havesup h ∈ (0 ,h ] sup H ∞ {| u h | + k X j =0 | u ( j ) | + | r h |} ≤ N k f k m (3.9)with a constant N depending only on d , m , δ, c , τ , K , M ,..., M m , | Λ | and k Λ k . Multiplying both sides of equation (3.8) by e − νt and then integratingthem over [0 , ∞ ) with respect to dt , we get expansion (3.6) with R h ( x ) := Z ∞ e − νt r h ( t, x ) dt,v ( j ) ( x ) := Z ∞ e − νt u ( j ) ( t, x ) dt, for j = 0 , . . . , k. Clearly, (3.9) implies that (3.7) holds with N depending only on d , m , δ, c , τ , K , M ,..., M m , | Λ | , and k Λ k . As we know the function u (0) in(3.8) is the B m solution of ∂∂t u = ( L + ν ) u ( t, x ) ∈ H ∞ ,u (0 , x ) = f ( x ) x ∈ R d , CCELERATED SCHEMES 17 which as we have seen in the proof of Theorem 3.2 guarantees that v (0) isthe unique B m solution of equation (3.2). (cid:3) Remark . We can show similarly that v ( i ) , i = 1 , ..., k , is the uniquesolution of the system L v ( j ) ( s, x ) + j X i =1 C ij L ( i ) v ( j − i ) = 0in an appropriate class of functions (cf. Theorem 2.2).The following result can be obtained easily from Theorem 2.7 by inspect-ing the proof of the previous theorem. Theorem 3.6.
Let p λ , q λ , c , and f satisfy the conditions of Theorem 3.5,with m − l in place of m in each of the conditions (i)–(iii) for an integer l ∈ [0 , m ] . Then D α v h is a bounded continuous function on R d for every multi-index α , | α | ≤ l , and the expansion (3.6) is valid with D α v h , { D α v ( j ) } kj =0 and D α R h in place of v h , { v ( j ) } kj =0 and R h , respectively. Furthermore, (3.7) holds with D α R h in place of R h and a constant N depending only on d , m , δ, c , τ , K , | Λ | , k Λ k , M , . . . , M m . In case (iii) we have v ( j ) = 0 for allodd j in the expansion. Set ¯ v h = k X j =0 b j v − j h , ˜ v h = ˜ k X j =0 ˜ b j v − j h , where ( b , b , . . . , b k ) and ˜ k , (˜ b , ˜ b , . . . , ˜ b ˜ k ) are defined in (2.22) and in (2.25).Then we have the following corollary. Corollary 3.7.
Suppose that the assumptions of Theorem 3.6 are satisfied.Then for every multi-index α with | α | ≤ l , sup R d | D α ¯ v h − D α v (0) | ≤ N k f k m h k +1 , and if condition (iii) is met then sup R d | D α ˜ v h − D α v (0) | ≤ N k f k m h k +1 , where N depends only on on d , m , δ , K , τ , c , | Λ | , k Λ k , M , . . . , M m . Proof of uniqueness in Theorem 2.1 and a stipulation
We will see later that the proof of Theorem 2.3 only uses the existenceof sufficiently smooth solutions of (2.8) and (2.13). Therefore, if m ≥ u (0) follows from expansion (2.18). If m = 2, one can usesimple ideas based on integrating by parts. We briefly outline these ideasreferring for details to [12], [13], [14].First, one may assume that g = f = 0 and let u (0) be the correspondingsolution. Then, by introducing a new function v = u (0) (cosh | x | ) − one reduces the issue to uniqueness of v , which satisfies an equation similar to(2.5) with g = f = 0 and different coefficients which we denote by ˆ q λ , ˆ p λ ,and ˆ c = c , and, moreover, v, Dv, D v ∈ L ( H T ). After that one multipliesthe equation for v by v and integrates over H T . One uses integration byparts, and the fact that due to the assumption q λ ≥ | D ˆ q λ | ≤ q λ sup | D ˆ q λ | . One also uses Young’s inequality implying that | v ( ∂ λ ˆ q λ ) ∂ λ v | ≤ N | v ˆ q / λ ∂ λ v | ≤ ˆ q λ ( ∂ λ v ) + N v , and the fact that 2ˆ v ˆ p λ ∂ λ ˆ v = ˆ p λ ∂ λ (ˆ v ) . Then one quickly arrives at a relationlike Z H T ( N − c ) | v | dxdt ≥ Z R d | v ( T, x ) | dx ≥ , where N is a constant independent of c . If c is large enough, the aboveinequality is only possible if v = 0, which proves uniqueness if c is largeenough. In the general case it only remains to observe that the usual changeof the unknown function taking v ( t, x ) e λt in place of v for an appropriate λ will lead to as large c as we like. Remark . Notice that apart from uniqueness in Theorems 2.1 and 2.2all our other assertions and assumptions are stable under applying molli-fications of the data with respect to x . For instance, take a nonnegative ζ ∈ C ∞ ( R d ) with unit integral, for ε > ζ ε ( x ) = ε − d ζ ( x/ε ) and forlocally summable ψ ( x ) use the notation ψ ( ε ) = ψ ∗ ζ ε . Then q ( ε ) λ , p ( ε ) λ , c ( ε ) , f ( ε ) , and g ( ε ) will satisfy the same assumptions with thesame constants as the original ones and will be infinitely differentiable in x .It is not hard to see that if our assertions are true for the mollified data,then they are also true for the original ones. For instance, let v ε be thesolution of (2.5) with the new data. The uniform in ε estimates of thederivatives in x and the equation itself, guaranteeing that the first derivativesin time are bounded, show that v ε are uniformly continuous in [0 , T ] × {| x | ≤ R } for any R . Then there is a sequence ε n ↓ v ε n convergesuniformly in [0 , T ] ×{| x | ≤ R } for any R to a bounded continuous function v .This along with uniform boundedness of | D α v ε | , | α | ≤ m , lead to the factthat the generalized derivatives | D α v | , | α | ≤ m , are bounded and admit thesame estimates as those of v ε . Also since D α v ε n → D α v in the sense ofdistributions and all of them are uniformly bounded, we conclude that thisconvergence is true in the weak sense in any L ([0 , T ] × {| x | ≤ R } ). Now itis easy to pass to the limit in equation (2.9) written for modified coefficientsand v ε in place of u concluding that since the derivatives converge weaklyand q ( ε ) λ → q λ ,..., f ( ε ) → f uniformly on H T , v satisfies (2.9).Similar argument takes care of Theorem 2.2 (in which uniqueness will bederived from uniqueness in Theorem 2.1).Our claim about stability of other results is almost obvious and CCELERATED SCHEMES 19 from this moment on we will assume that the data are as smooth in x as welike. Proof of Theorems 2.9 and 2.11
In [5] (see there Theorems 2.3 and 2.1 and Corollary 3.2 if m = 0) and[6] we obtained the following result on the smoothness in x of the solution u h to equation (2.1). Theorem 5.1.
Suppose that Assumptions 2.1 and 2.4 are satisfied. Supposethat (i) if m = 1 , then Assumptions 2.2 and 2.5 are satisfied, and (ii) if m ≥ , then Assumptions 2.2, 2.5, 2.6, and 2.7 are satisfied. Then for h ∈ (0 , h ] we have that D k u h , k = 0 , ..., m , are continuous in x and sup H T m X k =0 | D k u h | ≤ N ( F m + G m ) , (5.1) where F n = X k ≤ n sup H T | D k f h | , G n = X k ≤ n sup R d | D k g h | , and N depends only on τ , m , δ , c , K , | Λ | , k Λ k , M , ..., M m ( N dependson fewer parameters if m ≤ ). To proceed further we need a few formulas.
Lemma 5.2.
Let ϕ be a function on H T and n ≥ be an integer.(i) Assume that the derivatives of ϕ in x ∈ R d up to order n + 1 arecontinuous functions in x . Then for each h > D nh X λ ∈ Λ p λ δ h,λ ϕ = X λ ∈ Λ p λ Z θ n ∂ n +1 λ ϕ ( t, x + hθλ ) dθ (5.2) on H T , where ∂ λ ϕ is introduced in (2.12) .(ii) Assume that the derivatives of ϕ in x up to order n + 2 are continuousfunctions in x , and that Assumption 2.3 holds. Then D nh X λ ∈ Λ h − q λ δ h,λ ϕ = X λ ∈ Λ q λ Z (1 − θ ) θ n ∂ n +2 λ ϕ ( t, x + hθλ ) dθ, (5.3) on H T .Proof. By Taylor’s formula applied to ϕ ( t, x + hθλ ) as a function of θ ∈ [0 , δ h,λ ϕ ( t, x ) = Z ∂ λ ϕ ( t, x + hθλ ) dθ and δ h,λ ϕ ( t, x ) = ∂ λ ϕ ( t, x ) + h Z (1 − θ ) ∂ λ ϕ ( t, x + hθλ ) dθ. Multiplying the first equality by p λ and summing up in λ over Λ we obtain(5.2) for n = 0. Multiplying the second equality by q λ , summing up in λ over Λ we obtain (5.3) for n = 0 since X λ ∈ Λ q λ ∂ λ ϕ = 0due to Assumption 2.3.After that it only remains to differentiate n times in h both parts of theparticular case of formulas (5.2) and (5.3). The lemma is proved. (cid:3) Introduce u ( j ) h = D jh u h and observe that by Remark 2.6 under Assumption 2.1 the functions ∂ nλ u j are well defined if n + j ≤ m . By combining this with Lemma 5.2 and theLeibnitz formula we obtain the following. Corollary 5.3.
Let Assumptions 2.1 and 2.3 be satisfied. Let k ≥ be aninteger such that k + 2 ≤ m . Then u ( k ) h ( t, x ) = Z t (cid:0) L h u ( k ) h ( s, x ) + R kh ( s, x ) (cid:1) ds (5.4) on (0 , h ] × H T , where R kh ( t, x ) = k X i =1 C ik X λ ∈ Λ Z θ i (cid:2) p λ ( t, x )( ∂ i +1 λ u ( k − i ) h )( t, x + hθλ )+(1 − θ ) q λ ( t, x )( ∂ i +2 λ u ( k − i ) h )( t, x + hθλ ) (cid:3) dθ. Now we are ready to prove Theorems 2.9 and 2.11.
Proof of Theorem 2.9 . If m = 2 or k = 0, our assertion follow directlyfrom Theorem 5.1. Therefore, in the rest of the proof we assume that m ≥ k ≥ ≤ i ≤ k , then( i + 2) + r + ( k − i ) = k + 2 + r ≤ k + r ≤ m. Thus by Remark 2.6 we know that D i +2+ r u ( k − i ) h are bounded and continuouson H T . It follows that R kh ∈ B r . By Theorem 5.1 with r in place of m weobtain I kr := sup H T X j ≤ r | D j u ( k ) h | ≤ N sup H T X j ≤ r | D j R kh | . It is not hard to see that | D j R kh | ≤ N sup H T k X i =1 i +2+ j X n =1 | D n u ( k − i ) h | ≤ N k X i =1 I k − i,i +2+ j . CCELERATED SCHEMES 21
Hence, I kr ≤ N k X i =1 I k − i,i +2+ r . Here on the right the first index of I kr is reduced by at least 1 and the sumof indices increased by 2. Therefore, after k iterations we will come to theinequality I kr ≤ N I ,k +2 k + r . It only remains to observe that I , k + r ≤ I ,m and the latter quantity isestimated in Theorem 5.1. The theorem is proved. Proof of Theorem 2.11 . First of all observe that the symmetry assumptionand (2.16) imply that for any smooth function ϕ ( x ), odd i ≥
0, and anymulti-index α , such that | α | ≤ m , we have X λ ∈ Λ ( D α p λ ) ∂ i +1 λ ϕ = X λ ∈ Λ ( D α q λ ) ∂ i +2 λ ϕ = 0 . (5.5)If k = 1 and an integer n ≤ r , then owing to (5.5) (cid:12)(cid:12) D n X λ ∈ Λ q λ ( t, x )( ∂ λ u h )( t, x + hθλ ) (cid:12)(cid:12) = (cid:12)(cid:12) D n X λ ∈ Λ q λ ( t, x ) (cid:2) ( ∂ λ u h )( t, x + hθλ ) − ∂ λ u h ( t, x ) (cid:3)(cid:12)(cid:12) ≤ N h sup H T X i ≤ r | D i +4 u h | ≤ N h k u k m ≤ N ( k f k m + k g k m ) h =: N J h, where the last two estimates follow from the fact that r + 4 = r + 3 k + 1 ≤ m and from Theorem 2.9, respectively. Similarly, (cid:12)(cid:12) D n X λ ∈ Λ p λ ( t, x )( ∂ λ u h )( t, x + hθλ ) (cid:12)(cid:12) = (cid:12)(cid:12) D n X λ ∈ Λ p λ ( t, x ) (cid:2) ( ∂ λ u h )( t, x + hθλ ) − ∂ λ u h ( t, x ) (cid:3)(cid:12)(cid:12) ≤ N J h.
Hence, sup H T X n ≤ r | D n R h | ≤ N ( k f k m + k g k m ) h ≤ N J h and applying Theorem 5.1 to (5.4) yields (2.34).Now we proceed by induction on k . Assume that for an odd number j estimate (2.34) holds whenever 3 k + r ≤ m − and odd k ≤ j . Thishypothesis is justified by the above for j = 1 and to prove the theorem itsuffices to show that the hypothesis also holds with j + 2 in place of j . Takean odd k and an integer r such that k ≤ j + 2 , k + r ≤ m − and again use (5.4). As above, to obtain (2.34) it suffices to prove thatsup H T X n ≤ r | D n R kh | ≤ N J h. (5.6)Take an integer n ≤ r . Observe that if 1 ≤ i ≤ k and i is even, then k − i isodd and k − i ≤ j + 2 − i ≤ j and3( k − i ) + i + 2 + n = 3 k + n − i + 2 ≤ m − − i + 2 ≤ m − H T (cid:12)(cid:12) D n X λ ∈ Λ q λ ( t, x )( ∂ i +2 λ u ( k − i ) h )( t, x + hθλ ) (cid:12)(cid:12) ≤ N J h. (5.7)If 1 ≤ i ≤ k and i is odd, then i + 2 is odd too and as in the beginning ofthe proof (cid:12)(cid:12) D n X λ ∈ Λ q λ ( t, x )( ∂ i +2 λ u ( k − i ) h )( t, x + hθλ ) (cid:12)(cid:12) = (cid:12)(cid:12) D n X λ ∈ Λ q λ ( t, x ) (cid:2) ( ∂ i +2 λ u ( k − i ) h )( t, x + hθλ ) − ∂ i +2 λ u ( k − i ) h ( t, x ) (cid:3)(cid:12)(cid:12) ≤ N h sup H T X i ≤ k,l ≤ r | D l + i +3 u ( k − i ) h | , where the last sup is majorated by N J owing to Theorem 2.9 since3( k − i ) + r + i + 3 ≤ m − − i + 3 ≤ m. In both situations we have (5.7). Similarly, if 1 ≤ i ≤ k and i is odd, then i + 1 is even and (cid:12)(cid:12) D n X λ ∈ Λ p λ ( t, x )( ∂ i +1 λ u ( k − i ) h )( t, x + hθλ ) (cid:12)(cid:12) = (cid:12)(cid:12) D n X λ ∈ Λ p λ ( t, x ) (cid:2) ( ∂ i +1 λ u ( k − i ) h )( t, x + hθλ ) − ∂ i +1 λ u ( k − i ) h ( t, x ) (cid:3)(cid:12)(cid:12) ≤ N h sup H T X i ≤ k,l ≤ r | D l + i +2 u ( k − i ) h | , where the last sup is majorated by N J again owing to Theorem 2.9 since3( k − i ) + r + i + 2 ≤ m − − i + 2 ≤ m. Finally, if 1 ≤ i ≤ k and i is even, then k − i is odd, k − i ≤ j + 2 − i ≤ j ,and 3( k − i ) + r + i + 1 ≤ m − − i + 1 < m − , so that by the induction hypothesis (cid:12)(cid:12) D n X λ ∈ Λ p λ ( t, x )( ∂ i +1 λ u ( k − i ) h )( t, x + hθλ ) (cid:12)(cid:12) ≤ N J h, which is now shown to hold in both subcases. By combining this with (5.7)we come to (5.6) and the theorem is proved.
CCELERATED SCHEMES 23 Proof of Theorems 2.1, 2.2, and 2.10
Proof of Theorem 2.1.
First we replace q λ with symmetric ones using thefact that the symmetrization does not affect formula (2.7). To this endintroduce Λ s = Λ ∩ ( − Λ ) , ˆΛ = Λ ∪ ( − Λ ) . On Λ s we set ˆ q ± λ = (1 / q λ + q − λ ). If λ ∈ ± (Λ \ Λ s ) we set ˆ q λ = (1 / q ± λ .Then ˆΛ and ˆ q λ satisfy the symmetry condition (S) and can be used torepresent the first term on the right in (2.7) in place of the original ones.Next, we redefine and extend p λ introducing ˆ p λ on ˆΛ , so that ˆ p λ = M + p λ on Λ s , for λ ∈ Λ \ Λ s we set ˆ p ± λ = M ± (1 / p λ , and for − λ ∈ Λ \ Λ s we set ˆ p ± λ = M ∓ (1 / p − λ . (Remember that for the constant M fromAssumption 2.1 we have | p λ | ≤ M .) Then ˆΛ and ˆ p λ can be used torepresent the second term on the right in (2.7) in place of the original ones.One of the advantages of the new ˆ p λ is that ˆ p λ ≥
0, which implies that thenew χ λ satisfies Assumption 2.2.Define τ λ > δ ∈ (0 , δ = 1 /
2, if c is sufficiently large (independently of h ) and τ > , K , and K are chosen appropriately and depending only on d, | Λ | , k Λ k , M , M , M .We first concentrate on the case that c is indeed sufficiently large. In thatcase by Theorem 2.9, for h ∈ (0 , h ], there exists a unique solution u h ( t, x )of class B mT satisfying equation (2.1) with ˆ L h in place of L h , where ˆ L h isconstructed from ˆΛ , ˆ q λ , and ˆ p λ . Furthermore, k u h k m ≤ N ( k f k m + k g k m ) , (6.1)where N is a constant depending only on m, inf c , | Λ | , M ,..., M m , and k Λ k . Upon observing that owing to Remark 2.2 | ˆ L h u h | ≤ N (sup H T | D u h | + sup H T | Du h | + sup H T | u h | )with N independent of h , we conclude from the equation for u h that theirfirst derivatives in t are bounded uniformly in h . Therefore, there exists asequence h ( n ) ↓ u h ( n ) converges uniformly on [0 , T ] × { x : | x | ≤ R } for any R to a continuous function v . Then (6.1) implies that v ∈ B m and k v k m ≤ N ( k f k m + k g k m ) (6.2)with the same N as in (6.1). If we take τ λ ≡
1, then Remark 6.4 of [5] andRemark 4.3 of [6] imply that both N ’s can be chosen to depend only on d , m , inf c , | Λ | , and M , ..., M m .Next, the modified equation (2.1) yields that for any φ ∈ C ∞ ( R d ) and t ∈ [0 , T ] Z R d u h ( t, x ) φ ( x ) dx = Z R d g ( x ) φ ( x ) dx + Z t Z R d X λ ∈ ˆΛ u h ( s, x )[(1 / h,λ (ˆ q λ φ ) + δ h, − λ (ˆ p λ φ )]( s, x ) dxds + Z t Z R d ( − cu h + f ) φ ( s, x ) dx ds. We pass to the limit in this equation and find that v satisfies an integralequation, integrating by parts in which proves that v is a solution of (2.8).Finally, we notice that the case that c is not large is reduced to the aboveone by usual change of the unknown function taking v ( t, x ) e λt in place of v for an appropriate λ , which leads to subtracting λv from the right-hand sideof (2.5). For the new equation we then find a solution admitting estimate(6.2) with N independent of T but coming back to the solution of the originalequation will bring an exponential factor depending on T .This and uniqueness proved in Section 4 finish proving the theorem. (cid:3) Remark . In the above proof we considered arbitrary τ λ > m ≥
2, thenby Theorem 2.9 estimate (6.1) and hence (6.2) hold with N depending onlyon m, δ, c , τ , K , M , ..., M m , | Λ | , and k Λ k . This proves the assertion ofTheorem 2.10 regarding the constant N in Theorem 2.1. Proof of Theorem 2.2.
Notice that for each j = 1 , . . . , k equation (2.13) doesnot involve the unknown functions u ( l ) with indices l > j . Therefore we cansolve (2.13) and prove the statements (i) and (ii) recursively on j .First we prove that there is at most one solution ( u (1) , . . . , u ( k ) ) in thespace B × · · · × B . Denote S j = j X i =1 C ij L ( i ) u ( j − i ) . We may assume that u (0) = 0. Then clearly S = 0 and by Theorem 2.1we have u (1) = 0. If for a j ∈ { , . . . k } we have u (1) = u (2) = · · · = u ( j − = 0, then clearly S j = 0 which by Theorem 2.1 yields u ( j ) = 0. Hencethe statements on uniqueness follow because for every j = 1 , , . . . , k weobviously have B m − j ⊂ B when m ≥ k + 2 and B m − j ⊂ B when m ≥ k + 2.While dealing with the existence of a solution first take j = 1. Observethat by Theorem 2.1 we have u (0) ∈ B m with m ≥ m ≥ S ∈ B m − ⊂ B and byTheorem 2.1 it follows that there exists u (1) ∈ B m − satisfying (2.13) andadmitting the estimate k u (1) k m − ≤ N k u (0) k m . Taking the estimate of the last term again from Theorem 2.1 we obtain(2.14) for j = 1. In case (ii) we have actually better smoothness of S ,because the first sum in (2.11) is zero for i = 1 and, for that matter, for CCELERATED SCHEMES 25 all odd i . It follows that S ∈ B m − and this leads to (2.15) for j = 1 asabove. By adding that under the conditions (S) and (2.16) we have L (1) = 0, S = 0, and u (1) = 0, we obtain (2.17) for j = 1.Passing to higher j we assume that k ≥
2. Suppose that, for a j ∈{ , ..., k } we have found u (1) ,..., u ( j − with the asserted properties. Then inthe case (i) we have L ( i ) u ( j − i ) ∈ B m − j ⊂ B for i = 1 , . . . , j , since m − j − i ) − ( i + 2) = m − j + 2 i − ≥ m − j ≥ . Hence S j ∈ B m − j and therefore by Theorem 2.1 there exists u ( j ) ∈ B m − j satisfying (2.13) and admitting the estimate k u ( j ) k m − j ≤ N j X i =1 k u ( j − i ) k m − j +3 ≤ N j X i =1 k u ( j − i ) k m − j − i ) ≤ N ( k f k m + k g k m ) , where the last inequality follows by the induction hypothesis.In case (ii) we take into account that due to condition (S) we have X λ ∈ Λ q λ ∂ i +2 λ ϕ = 0 , (6.3)and due to condition (2.16) we have X λ ∈ Λ p λ ∂ i +1 λ ϕ = 0 (6.4)for odd numbers i and sufficiently smooth functions ϕ . It follows that incase (ii) for i = 1 , ..., j we have L ( i ) u ( j − i ) ∈ B m − j ⊂ B , since L (1) u ( j − ∈ B m − j − − and for i ≥ m − j − i ) − ( i + 2) = m − j + i − ≥ m − j ≥ . Hence S j ∈ B m − j and therefore by Theorem 2.1 there exists u ( j ) ∈ B m − j satisfying (2.13) and admitting the estimate k u ( j ) k m − j ≤ N k u ( j − k m − j +2 + N j X i =2 k u ( j − i ) k m − j +3 ≤ N j X i =1 k u ( j − i ) k m − j − i ) , and by using the induction hypothesis we come to (2.15).Furthermore, in case (ii) if (2.16) is satisfied, our induction hypothesissays that u ( l ) = 0 for all odd l ≤ j −
1. If j is even, then, obviously, u ( l ) = 0 for all odd l ≤ j as well. If j is odd then to carry the induction forward itonly remains to prove that u ( j ) = 0. However, for odd i we have L ( i ) u ( j − i ) = 0due to (6.3)-(6.4). This equality also holds if i ≥ i is even, since then j − i is odd and u ( j − i ) = 0 by assumption. Thus, S j = 0 and u ( j ) = 0. (cid:3) Remark . The above proof is based on Theorem 2.1 and leads to esti-mates (2.14) and (2.15) with N depending only on the same parametersas in Theorem 2.1. Therefore, according to Remark 6.1 if Assumptions 2.1through 2.6 are satisfied and the restrictions on m and k from Theorem 2.2are met, then the constants N in estimates (2.14) and (2.15) depend onlyon m, δ, c , τ , K , M , ..., M m , | Λ | , and k Λ k . This proves the part of asser-tions of Theorem 2.10 concerning Theorem 2.2. The proof of its remainingassertions can be obtained in the same way and is left to the reader.7. Proof of Theorem 2.3 and 2.7
We need some lemmas. The first one is a simple lemma from undergrad-uate calculus on Taylor’s expansion.
Lemma 7.1.
Let F be a real-valued function on (0 , such that for aninteger m ≥ the derivative F ( m +1) ( h ) of order m +1 exists for all h ∈ (0 , ,and F ( m +1) is a bounded function on (0 , . Then F ( k ) (0) := lim s ↓ F ( k ) ( s ) exist for ≤ k ≤ m , and F ( h ) = m X k =0 h k k ! F ( k ) (0) + R m ( h ) holds for h ∈ [0 , with R m ( h ) = Z h ( h − s ) m m ! F ( m +1) ( s ) ds, so that | R m ( h ) | ≤ sup s ∈ (0 , | F ( m +1) ( s ) | h m +1 ( m + 1)! for all h ∈ [0 , . To formulate our next lemma we recall the operators L h , L and L ( i ) ,defined in (2.2), (2.7), and (2.11), respectively, and for each h ∈ (0 , h ] andinteger j ≥ O ( j ) h = L h − L − X ≤ i ≤ j h i i ! L ( i ) . CCELERATED SCHEMES 27
Lemma 7.2.
Let Assumption 2.3 hold. Assume that for some integer l ≥ the functions p λ , q λ belong to B l for all λ ∈ Λ . Then for any integer j ≥ kO ( j ) h ϕ k l ≤ N k ϕ k l + j +3 h j +1 (7.1) for all h ∈ (0 , h ] and ϕ ∈ B l + j +3 , where N is a constant depending onlyon | Λ | , M , ..., M l .Proof. We may assume that the derivatives in x of ϕ up to order l + j + 3are bounded continuous functions on H T . By Lemma 5.2 the derivatives ofthe function L h ϕ in h up to the ( l + j + 1)st order are bounded functionson (0 , h ] × H T and ( L φ )( t, x ) = lim h → ( L h ϕ )( t, x ) , ( L ( i ) φ )( t, x ) = lim h → ( D ih L h φ )( t, x ) . Thus applying Lemma 7.1 to F ( h ) := L h ϕ ( t, x ) for fixed ( t, x ) and usingLemma 5.2, we have O ( j ) h ϕ = Z h ( h − ϑ ) j j ! L ( j +1) ϑ ϕ dϑ = X λ ∈ Λ q λ Z h ( h − ϑ ) j j ! Z (1 − θ ) θ j +1 ∂ j +3 λ ϕ ( t, x + ϑθλ ) dθ dϑ + X λ ∈ Λ p λ Z h ( h − ϑ ) j j ! Z θ j +1 ∂ j +2 λ ϕ ( t, x + ϑθλ ) dθ dϑ. Now estimate (7.1) follows easily. (cid:3)
The next lemma is a version of the maximum principle for ∂/∂t − L h . Itis a special case of Corollary 3.2 in [5]. Lemma 7.3.
Let Assumption 2.1 with m = 0 be satisfied and let χ h,λ ≥ for all λ ∈ Λ . Let v be a bounded function on H T , such that the partial de-rivative ∂v ( t, x ) /∂t exists in H T . Let F be a nonnegative integrable functionon [0 , T ] , and let C be a nonnegative bounded function on H T such that ν := sup H T ( C − c ) < . Assume that for all ( t, x ) ∈ H T we have ∂∂t v ≤ L h v + C ¯ v + + F, (7.2) where ¯ v ( t ) = sup { v ( t, x ) : x ∈ R d } . Then in [0 , T ] we have ¯ v ( t ) ≤ ¯ v + (0) + | ν | − sup [0 ,t ] F, (7.3) where a + := ( | a | + a ) / for real numbers a . Proof of Theorem 2.3.
By taking u h e − ( M +1) t in place of u h , we may assumethat c ≥
1. Consider first the case k = 0. Since m ≥
3, by Theorem 2.1equation (2.7) has a solution u (0) , which belongs to B m and estimate (2.10)holds. Clearly, w := u h − u (0) is the unique bounded solution of the equation w ( t, x ) = Z t (cid:0) L h w ( s, x ) + F ( s, x ) (cid:1) ds, ( t, x ) ∈ H T , (7.4)where F := O (0) h u (0) = L h u (0) − L u (0) . By Lemma 7.2 and estimate (2.10) kO (0) h u (0) k ≤ N X λ ∈ Λ ( k p λ k + k q λ k ) k u (0) k h ≤ N ( k f k + k g k ) h with constants N depending only on d , | Λ | M , M , M , and T . Afterthat an application of Lemma 7.3 to equation (7.4) proves the statement ofTheorem 2.3 for k = 0.Let k ≥
1. Then by Theorem 2.2 the system of equations (2.13) has abounded solution { u ( i ) } ki =1 . Observe that for w := u h − k X j =0 u ( j ) h j j ! (7.5)we have equation (7.4) with F := L h u (0) − L u (0) + k X j =1 L h u ( j ) h j j ! − k X j =1 L u ( j ) h j j ! − G, and G := k X j =1 j X i =1 i !( j − i )! L ( i ) u ( j − i ) h j = k X i =1 k X j = i i !( j − i )! L ( i ) u ( j − i ) h j = k X i =1 k − i X l =0 i ! l ! L ( i ) u ( l ) h l + i = k − X l =0 h l l ! k − l X i =1 h i i ! L ( i ) u ( l ) = k X j =0 h j j ! X ≤ i ≤ k − j h i i ! L ( i ) u ( j ) . Hence by simple arithmetics F = k X j =0 h j j ! O ( k − j ) h u ( j ) . (7.6)Notice that k − j + 3 ≤ m − j for j = 0 , , . . . , k in case (i) ,k − j + 3 ≤ m − j for j = 0 , , . . . , k in case (ii) ,k − j + 3 ≤ m − j for j = 0 , , . . . , k − . CCELERATED SCHEMES 29
Therefore by Theorem 2.2 under each of (i), (ii), and (iii) k u ( j ) k k − j +3 ≤ N ( k f k m + k g k m )for j = 0 , . . . , k ( u ( k ) = 0 in the case (iii)). Thus by Lemma 7.2 kO ( k − j ) h u ( j ) k ≤ N h k − j +1 k u ( j ) k k − j +3 ≤ N h k +1 − j ( k f k m + k g k m ) . Consequently, k F k ≤ N ( k f k m + k g k m ) h k +1 for h ∈ (0 , h ] , where N depends only on d , m , | Λ | , M , . . . , M m , and T . Hence we get(2.18) by Lemma 7.3, and the proof is complete. (cid:3) Proof of Theorem 2.7.
Coming back to the above proof of Theorem 2.3 wesee that function (7.5) satisfies (7.4) with F given by (7.6). We notice that k − j + 3 + l ≤ m − j for j = 0 , , . . . , k in case (i) ,k − j + 3 + l ≤ m − j for j = 0 , , . . . , k in case (ii) ,k − j + 3 + l ≤ m − j for j = 0 , , . . . , k − . Therefore by Theorem 2.1, when k = 0, and by Theorem 2.2, when k ≥ k u ( j ) k k − j +3+ l ≤ N ( k f k m + k g k m )for j = 0 , . . . , k ( u ( k ) = 0 in case (iii)). By Theorem 2.10 the constant N depends only on m , δ , c , τ , K , M , ..., M m , | Λ | , and k Λ k . By Lemma7.2 kO ( k − j ) h u ( j ) k l ≤ N h k − j +1 k u ( j ) k k − j + l +3 , where N is a constant depending only on | Λ | , M ,. . . M l . Hence k F k l ≤ N ( k f k m + k g k m ) h k +1 for h ∈ (0 , h ] . Consequently, applying Theorem 2.9 to equation (7.4), for any multi-index α , | α | ≤ l , for r ( α ) h := h − ( k +1) (cid:0) D α u h − k X j =0 D α u ( j ) h j j ! (cid:1) we have k r ( α ) h k = h − ( k +1) k D α w k ≤ N ( k f k m + k g k m ) , with a constant N depending only on m , d , δ , c , τ , K , M , ..., M m , | Λ | and k Λ k , which proves the theorem. (cid:3) References [1] H. Blum, Q. Lin, and R. Rannacher,
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